Experimental Identification of Rear Wheel Slip Dynamics of a Motorbike

Proceedings of the 15th IFAC Symposium on System Identification Saint-Malo, France, July 6-8, 2009

Experimental Identification of Rear Wheel Slip Dynamics of a Motorbike Matteo Corno and Sergio M. Savaresi ∗ ∗

M. Corno and S. M. Savaresi are with Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci, 32, 20133 Milano. ITALY {corno, savaresi}@elet.polimi.it

Abstract: This paper describes the black-box identification of the rear wheel slip dynamics of a racing motorbike equipped with an electronic throttle. The system’s response to two different inputs is studied: throttle and spark advance. After the description of the experimental setup, issues regarding signal processing and data analysis are discussed. Two types of tests, step and frequency sweep responses, are performed to evaluate the differences between the available actuation methods. The experimental data analysis allows to derive a model useful for traction control design and shows that throttle action is only marginally slower than spark advance. 1. INTRODUCTION In the past several years, the automotive market have witnessed the mass diffusion of closed-loop control systems like anti-locking systems (ABS), active stability (ESP) and traction control. The racing track has been the cradle of most of these technologies. Two-wheeled vehicles are following the same path, but they have fallen behind in the diffusion of these technologies to the general public. The reasons of this lagging are two-fold: cultural and technological. Sport motorcycles are considered recreational vehicles and motorcyclists do not like any intervention that they think would make the motorcycle easier to ride. On the other hand motorcycles exhibit more complex dynamics than four-wheeled vehicles. Although the first scientific papers on bicycles and their stability can be traced back to end of the 19th century, only in the past years the complexity involved in modeling and simulation of two-wheeled vehicles could be tackled thanks to the use of multi-body approach. Modern multi-body tools allow for reliable and realistic modeling and simulation of twowheeled vehicles. Many authors have developed simulators (see for example Sharp et al. [2004], Cossalter and Lot [2002], Ferretti et al. [2006]) and studied motorbike stability under different driving conditions (Sharp [1971], Cossalter et al. [2002]). Multi-body simulators are well suited for dynamic modal analysis, but they have two disadvantages that render them unsuited for control system design purposes. Firstly, they are often too complex for model based control system design and thus require order reduction techniques (as shown in Corno et al. [2008a, 2007]) and, secondly, the great number of parameters that characterizes a motorcycle makes their validation lengthy and expensive. The main contribution of this paper is the identification of the rear wheel slip dynamics of a racing motorbike during ⋆ This work has been supported by MIUR project New methods for Identification and Adaptive Control for Industrial Systems.

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straight running. The employed identification protocol has the advantage of being fast and requiring only a patch of straight track. The black-box identification approach (see Ljung [1999]) allows to focus on the relevant dynamics for traction control purposes, thus surmounting the hurdle of parameter identification. The proposed protocol allows to shed some light on a long standing questions among racing engineers: is slip control via throttle actuation feasible? In order to answer this question, two models of the slip dynamics are identified, one from throttle position and the second from spark advance. To the best of the Authors’ knowledge, no previous result on the specific problem of rear wheel slip dynamics analysis for two-wheeled vehicles is available in the open scientific literature. The paper is organized as follows: in Section 2 the experimental setup is presented along with signal processing and filtering; Section 3 is devoted to the identification of the slip dynamics from throttle position and spark advance. The two dynamics are finally compared in Section 4. 2. EXPERIMENTAL SETUP In the present work the rear wheel slip dynamics of a racing motorbike is studied. The motorbike is propelled by a 1000cc 4-stroke engine; it weights about 160 kg (without rider) and can deliver more than 200 HP. For confidentiality reasons other details of the motorbike are kept undisclosed as well as some plots normalized. In addition, the vehicle is equipped with (1) an electronic throttle body which allows to electronically control the position of the throttle valve independently from the rider’s request. The electronic throttle body is controlled by a PID servo controller; its bandwidth will be referred to as ωc ; (2) an Electronic Control Unit (ECU) that allows to change the engine spark advance and control the throttle. The clock frequency of the ECU is 1Khz. (3) two wheel encoders to measure the wheels’ angular velocity;

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Fig. 2. Rear wheel velocity, rear suspension compression and engine RPM spectra (lower) - on the x axis the wheel rolling frequency harmonics are marked.

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Fig. 1. Fixed-position estimation error upper bound and delay as a function of motorcycle velocity for different encoder resolutions. (4) a 1-dimensional Corrsys-Datron optical velocity sensor. This sensor measures the true longitudinal velocity and it will be used to compute the instantaneous wheel slip. In the remainder of the section the additional set up is detailed. 2.1 Wheel Encoders The motorbike is equipped with two 16-tooth encoders to measure wheels’ angular velocity. They are made of two elements: a rotating part with teeth (or lines) and a fixed sensor which detects the passage of a line; the sensor outputs a pulse every time a line passes in front of it. Due to the widespread usage of these type of sensors, different algorithms for speed estimation have been developed (see e.g., Brown et al. [1992]). The choice of the algorithm is strongly application-dependent since no globally optimal algorithm exists (see Bascetta et al. [2007]). For this application, where high rotational velocities are expected, an algorithm based on the fixed position approach has been chosen. The speed is estimated as ω ˆ=

2π N

TT m

2π

=  TTN

∆t where TT m is the time interval between two successive ascending fronts as measured by the sensor fixed elements, and ∆t is the clock period of the employed microprocessor. The limiting factor is the microprocessor clock which determines the upper bound of the estimation error; in the present application the time resolution is ∆t = 1µs. Moreover, being the time between teeth measured asynchronously, the algorithm suffers from a velocity dependent estimation delay which can be written as: τ (ω) = TT m (ω). Fig. 1 shows the estimation error and delay for different velocities. It is clear that the estimation delay becomes critical at low speed when the time between teeth is longer. The figure shows that with 16 teeth, when the velocity drops below 40 km/h the estimation delay is more than 10 ms; this translates in a phase loss of 36◦ at 10Hz. It is therefore clear that for the current application, the choice ∆t

of the encoder resolution is critical. Increasing the number of teeth would guarantee a smaller estimation delay, causing only a marginal precision degradation, thanks to the high time resolution of the ECU. 2.2 Signal Processing By inspecting the wheel velocity signals at constant speed, it can be noted that they exhibit a periodic component. A better insight of the problem is gained by looking at the spectrum of the rear wheel velocity signal, as represented in Fig. 2. The disturbance that affects the measurement corresponds to the wheel rolling frequency. It is speculated that the disturbance is caused by two factors: (1) the wheel is not perfectly balanced; this may be caused by uneven tread consumption or other asymmetries in the tire; (2) the wheel encoder is not perfectly aligned with the axis of the wheel; or the teeth of the wheel may not be uniformly distributed around the encoder. Assuming a wheel encoder with 10 cm radius, the noise can be explained by a 3 mm misplacement of one of the teeth. Seen the above considerations the harmonic component is treated as a measurement noise; although this noise does not represent an issue at high speed, when the 1X frequency is well above the slip dynamics frequency, it can fall in the slip dynamics bandwidth at low speed (when a slip control is most needed). In particular, the 1X frequency is below 10Hz for speeds slower than 70km/h. It is thus critical to address the measurement noise. This can be done via an adaptive notch filter (see Lim et al. [2005]). Fig. 3 depicts the filtering scheme employed in the analysis. After a first low pass filtering stage, the 1X component

Fig. 3. Time between teeth filtering scheme. is removed from the signal with a second order adaptive notch filter. The wheel frequency is estimated using a low pass filter with 1Hz cutoff frequency. Being the scope of this work dynamics identification, all the filtering is done

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the describing-function approach has been adopted (see e.g. Gelb and der Velde [1968]); by “slicing” the frequency sweep and by looking at a single frequency at a time, it is possible to extract information on amplification and phase shift of each harmonics via Fourier transform. It is assumed that if an input signal is considered, then the output signal can be written as: N X Ai (ω) sin(i ωt + ψi (ω)) (1) y(t) ≈

measured set point

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i=1

1

where Ai (ω) and ψi (ω) are the amplification and phase shift of the i-th harmonic and N is the number of harmonics that are taken into account. If N =1, a classical describing function is obtained. Fig. 5 graphically depicts the proposed model.

0 −1 −2 time

Fig. 4. Example of a throttle sweep experiment. Top: requested throttle position; bottom: measured wheel slip. off line. A slip controller had to be designed, the filtering issues would be more critical since it would not be possible to use acausal filters. 3. IDENTIFICATION OF ENGINE-TO-SLIP DYNAMICS

Fig. 5. Nonlinear slip dynamics model.

Among the different conceivable methods that can be employed to control the rear wheel slip during acceleration, the two most obvious are throttle and spark advance. By varying the throttle position or the spark advance, one can modulate the torque generated by the engine and consequently the wheel slip. The output variable is the longitudinal slip: ωr rr − v λ= v where ωr and r are the rotational speed and rolling radius of the rear wheel, respectively, whereas v is the longitudinal vehicle speed (measured with the optical sensor). The longitudinal slip of the rear wheel is the natural output variable, since the TC-system has the goal of regulating λ to a target value. In order to identify the slip dynamics, two kinds of tests have been devised: frequency sweep and step response. Both kinds of test are done on a 3.5 km straight dry asphalt patch; the rider is asked to bring the motorcycle to a given constant engine speed in a given gear (14000 RPM and 2nd gear in the shown experiments). After steady state condition is reached, the rider presses a button which commences the trial. The throttle control is taken over by the ECU and the excitation signal (a frequency sweep or a square wave) is applied around the neighborhood of the initial condition. Fig. 4 depicts the requested and actual throttle position and the resulting slip. The frequency sweep response is used as the principal signal for identification as it provides a clear picture of the Input/Output behavior. Notice that the overall I/O behavior is significantly nonlinear. Due to the highly nonlinear nature of the system the issue of the class of mathematical models to be employed for system identification arises. Considering that the model is control-oriented, an extension to higher harmonics of

The system is composed of four elements: an optional pure delay (which will be discussed later) and three harmonic generators. The first harmonic generator is a classical linear system with frequency response G1 (jω); the 2nd and 3rd harmonic generators are assumed to be nonlinear systems which, when fed with a sinusoidal input, generate a sinusoidal output at, respectively, twice and three times the input frequency; moreover the amplification and phase shift introduced are assumed to depend on the input’s frequency. Thus, for each input frequency ωi , it is possible to characterize the harmonic generators via a complex number which represents the amplification and phase shift: G1 (jωi ) =

Λy (j2ωi ) Λy (j3ωi ) Λy (jωi ) G2 (jωi ) = G3 (jωi ) = Λu (jωi ) Λu (jωi ) Λu (jωi ) (2)

where Λy and Λu are, respectively, the Fourier transform of the input and the output. G1 (jω), G2 (jω) and G3 (jω) (which can be regarded as higher order describing functions) can be computed numerically from the experiments and then represented in a Bode-like diagram. Fig. 6 depicts the amplification and phase shift of each harmonic, in the case of a throttle sweep experiment, as a function of the input frequency. The frequency in figure is normalized with respect to ωc . From Fig. 6 it is possible to draw the following remarks: (1) the first harmonic is the dominant one. Up to input frequencies of 0.25 ωc the magnitudes of the second and third harmonics are negligible, between 0.25 ωc and 0.7 ωc both the second and third harmonics have a significant energy contribution, while after 0.7 ωc the third harmonic is considerably damped. (2) All the three harmonics have a resonant behavior. In particular the first harmonic has a resonance at around 0.8 ωc , the second at 0.7 ωc and the third at 0.45 ωc . The resonant behavior of the system is attributed to the elasticity of the transmission. The describing functions can be characterized in a more compact way as generalized frequency responses via com-

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By looking at the time domain response, it is clear that the nonlinearity is due to the asymmetric behavior of the system (rising dynamics is different from falling dynamics). In order to take into account this issue and derive a model useful for control system design, a switching linear system is proposed. Two linear systems are defined and the switching between the two is driven by the throttle reference first derivative: x(t) ˙ = Ai x(t) + Bi u(t) y(t) = Cx(t)

st

1 harmonic 2nd harmonic rd

where

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1

Fig. 6. Describing functions up to the third harmonics of the throttle-to-slip dynamics and their rational approximations. The plots are reported from u to y, i.e. the pure delay is also represented in the phase. plex functions. The higher order describing functions of the first three harmonics can be described by rational complex functions of the form: b1 (jω)n + b2 (jω)n+1 + . . . bn+1 (jω) B(jω) = , G(jω) = A(jω) a1 (jω)m + a2 (jω)m−1 + . . . am+1 (jω) (3) where ω ∈ R. The parameters ai and bi are determined by solving the following optimization problem: 2 l X B(ω(k)) min wf (k) h(k) − , (4) b,a A(ω(k)) k=1

where l is the number of available frequencies, wf (k) is a weight that can be used to drive the fitting toward certain frequencies, h(k) is the experimental frequency response. In the present work the optimization problem has been solved via an iterative approach based on the damped Gauss-Newton method (see Dennis and Schnabel [1983]). The order of the numerator and denominator is determined with a method adapted from the classical Finite Prediction Error (Ljung [1999]) method, which allows to find a trade-off between model complexity and accuracy. Notice that the first harmonic generator, thanks to the linearity hypothesis, can be treated and analyzed as a transfer function and can be regarded as a linear approximation of the throttle-to-slip dynamic. As mentioned in Fig. 5, a 10 ms pure delay has been introduced to model the air-box dynamics. This hypothesis is confirmed also by the black-box approach; if the delay is not introduced the optimization problem (4) yields a non-minimum phase transfer function (for the first harmonic), which cannot be physically explained. The fitting between the experimental describing functions and the analytical expression in the frequency domain is shown in Fig. 6, as it can be seen the fitting is good both in magnitude and phase. The identified describing functions can also be validated in the time domain by comparing the measured slip at different frequencies with the output

where Aopen , Bopen , Aclose , Bclose and C are the observability canonical form realization of a linear system with the structure identified by the first harmonic approximation. The choice of the observability canonical form allows a smooth transition between the two systems simply by keeping the state vector unchanged at the switching. The parameters of the two linear models have been identified from the step response experiments; this kind of experiments allow to better isolate the asymmetries. The first harmonic approximation transfer function has been used as the initial point for the following non-convex optimization problem:  2 N X y(k) − ysim (µ, ωz , ωp ) wt(k) min (5) µ,ωz ,ωp N k=1

where y(k) is the measured slip, ysim (µ, ωz , ωp ) is the slip simulated with the switched system, N is the number of samples in the experiment and wt is a weight. Note that only the gain, real pole and real zero are subject to optimization. The resonant mode is not changed because it is attributed to the transmission chain dynamics whose dynamics is assumed symmetric. Fig. 8 shows the obtained fitting in the step reference experiments. 2

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Fig. 8. Measured (dotted) and simulated (solid) throttle step responses. The overall fitting both in the opening and closing experiments is satisfactory; especially if one considers that even before the reference is subjected to the step, there are fluctuations of 0.5% in the slip value. This conclusion is confirmed by the validation test run on the sweep data. Fig. 9 shows the response of the switched system for three

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Fig. 9. Validation of the switched system on a throttle sweep experiment. Detail at 0.3 ωc (left), at 0.5 ωc (center) and 0.9 ωc (right). The plots show the throttle input, the measured slip and two simulated slips (first harmonic approximation and switched system). 4. ANALYSIS OF ENGINE-TO-SLIP DYNAMICS

different frequencies. The switched system successfully captures the asymmetric behavior. The same identification procedure can be employed on the spark advance experiments yielding similarly good results. The identification of the spark to slip dynamics is summarized in Fig. 10. It is possible to draw the following magnitude [dB]

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Fig. 10. Describing functions up to the third harmonics of the spark advance-to-slip dynamics and their rational approximations. The plots are reported from u to y, i.e. the pure delay is also represented in the phase. remarks: (1) as expected, the first harmonic is the dominant one. The three zones identified for the throttle dynamics are still present, but the spark-to-slip dynamics is characterized by a faster rise of the second and third harmonics. (2) The three harmonics still show a resonant behavior. In particular the first harmonic has a resonance at around 0.8 ωc , the second at 0.6 ωc and the two resonances of the third harmonic are clearly visible at 0.35 ωc and 0.54 ωc .

Electronic throttle control in motorbikes have been only recently introduced (see Corno et al. [2008b]); before the introduction of such technology, engine torque was (and still is) mainly modulated via spark advance control. This method has some advantages and disadvantages over throttle regulation; in particular: (1) spark advance regulation is built-in in most engine control units and does not require any extra hardware nor the design of an additional control loop. (2) Spark advance regulation allows a faster control of the engine torque. The spark acts right before the torque is generated. In term of engine torque, spark advance effects are measured in the same engine cycle in which the variation is applied. (3) Optimal operation of the engine requires the spark advance to be in a relatively small range; if engine torque is regulated via the spark advance for a long period the spark advance may drift out of the optimal range and the engine efficiency may be affected. (4) Spark advance guarantees a smaller modulability range than throttle control, i.e. at a given engine speed the torque variation that can be generated by modifying spark advance is limited whereas throttle control can reach the engine torque limits. In this section the first harmonic approximations derived in the previous section are employed to draw conclusions relevant to the design of a traction control for racing motorbikes. In order to have a better vision of the features of the dynamics, the first harmonics approximation for the throttle-to-slip and spark-to-slip dynamics are reported in Fig. 11. From figure, it is possible to draw the following conclusions: (1) both dynamics show a resonance around 0.8 ωc . This resonance is due to the transmission. It is interesting to note that the resonance in the throttle-to-slip dynamics

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magnitude [dB]

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throttle−to−slip

−2 −4 −6 −8 0.01

spark−to−slip

0.1

phase [°]

0

1 spark−to−slip

−50 −100 −150 0.01

throttle−to−slip

60° 0.1 normalized frequency [ωc]

1

Fig. 11. spark advance-to-slip and throttle-to-slip dynamics. For easier comparison the low frequency gains are normalized to 1. is less damped than in the spark-to-slip; this is due to the throttle servo. As shown in Fig. 4 the actual throttle exhibits a resonance right before ωc ; this resonance is coupled with the intrinsic slip resonance. (2) As anticipated, spark advance is shown to be faster than throttle action. At ωc , there is a 60◦ difference in phase. This observation is the key conclusions of this paper. It proves that although torque control is better achieved by spark advance; the real bottleneck is the transmission of the torque from the engine to the tire and that slip control through throttle control is indeed achievable. A bandwidth of 0.7-0.8 ωc can be anticipated with throttle actuation and a bandwidth of 0.8-0.9 ωc with spark advance actuation. Only a marginal difference. (3) Although spark advance allows a slightly faster actuation, it should also pointed out that the response of the system to spark advance variation is less linear, and therefore more difficult to model and control. 5. CONCLUSIONS In this paper the rear wheel slip dynamics of a racing motorbike have been studied. An identification protocol which is based on frequency sweeps and step responses was proposed. The black-box approach allowed to derive a simple but complete model which can be employed in model-based traction control design. The proposed identification protocol, which is simple to implement, was employed to study the differences between slip actuation via throttle and spark advance control. It was shown that, although spark advance is the preferred way to control engine torque, when it comes to controlling slip, the transmission and tire dynamics make spark advance action only marginally faster than throttle action.

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These considerations open the way to mixed slip controllers; having two control variables allows to implement advanced strategies that, for example, achieve fine traction control without impeding engine efficiency.

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